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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > psubclssatN | Structured version Visualization version GIF version |
Description: A closed projective subspace is a set of atoms. (Contributed by NM, 25-Jan-2012.) (New usage is discouraged.) |
Ref | Expression |
---|---|
psubclssat.a | ⊢ 𝐴 = (Atoms‘𝐾) |
psubclssat.c | ⊢ 𝐶 = (PSubCl‘𝐾) |
Ref | Expression |
---|---|
psubclssatN | ⊢ ((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐶) → 𝑋 ⊆ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | psubclssat.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
2 | eqid 2733 | . . 3 ⊢ (⊥𝑃‘𝐾) = (⊥𝑃‘𝐾) | |
3 | psubclssat.c | . . 3 ⊢ 𝐶 = (PSubCl‘𝐾) | |
4 | 1, 2, 3 | psubcliN 38747 | . 2 ⊢ ((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐶) → (𝑋 ⊆ 𝐴 ∧ ((⊥𝑃‘𝐾)‘((⊥𝑃‘𝐾)‘𝑋)) = 𝑋)) |
5 | 4 | simpld 496 | 1 ⊢ ((𝐾 ∈ 𝐷 ∧ 𝑋 ∈ 𝐶) → 𝑋 ⊆ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ⊆ wss 3947 ‘cfv 6540 Atomscatm 38071 ⊥𝑃cpolN 38711 PSubClcpscN 38743 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-iota 6492 df-fun 6542 df-fv 6548 df-psubclN 38744 |
This theorem is referenced by: pmapidclN 38751 psubclinN 38757 paddatclN 38758 pclfinclN 38759 poml6N 38764 osumcllem3N 38767 osumcllem9N 38773 osumcllem11N 38775 osumclN 38776 |
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